204 IX ANALYTIC FUNCTIONS

An entire function of p variables is a mapping / of Kp into E which is
equal to the sum of a power series which is absolutely summable in the
whole space Kp (cf. (9.9.6)). For each b e Kp, f(z) is then equal to the sum
of a power series in the zk-bk, which is absolutely summable in the whole
space Kp by (9.3.1).

(9.3.2) Let A be an open subset 0/Kp, B an open subset ofKq, gk(l^k </?)
p scalar functions defined and analytic in B, and suppose the image of B by
>•*-, 9p) & contained in A. Then, for any analytic mapping f of A into E,
, • • • > ffp) is analytic in B.

This follows at once from the definition and from (9.2.2). In particular,
if /is analytic in AcKp, then, for any system (aq+l9 ...9ap) of p-q
scalars, (zl9 ...,%) -*f(zl9 ...9z9, aq+l9 ...9ap) is analytic in the open set
A(aq+l,...,ap) (3.20.12) in K*

(9.3.3) In order that a mapping /= (/19 ...9fq) of A cr Kp ITI^O K9 6^ analytic
in A, zY w necessary and sufficient that each of the scalar functions ft (1 < / ^ ^)
Ae analytic in A.

Obvious from the definition.

(9.3.4) Le* zfc = ^ + % for l^k^p, xk and yk being real Iff is analytic
in A c= Cp, then (x1? yl9..., xp, jp) -»/(X:L 4- z>i,..., xp + iyp) is analytic in A,
considered as an open set in R2p.

Indeed, that function is analytic in the open subset B c= C2p, inverse
image of A by the mapping (wl5 vl9..., up, vp) -»(WjL + ivi9..., up + ivp) of
C2p into Cp, by (9.3.2). Hence it is analytic in A = B n R2p, when A is
considered as a subset of R2p.

(9.3.5) Let (cnin2... rtpzn1i ... znpp) be a power series which is absolutely sum-
mable in an open poly disk P of center 0, and letf(z) be its sum. Then the power
series (nkcnin2... Bpz? ... zj*'1... zp) is absolutely summable in P and its sum
is the partial derivative Dkf (=df/dzk). — \bt\ (1 < i < p).